1. Field of the Invention
The present invention relates to a mesh generation method, and in particular to a mesh generation method employing bubbles.
2. Related Art
Meshing, or mesh generation, is a process for dividing a geometric model generated by CAD etc. into a set of small elements. Mainly, a triangle mesh and a quadrilateral mesh are used for a two-dimensional mesh. In a computer simulation, such as an analysis of a car collision, since a reliable solution can not be obtained by using a triangle mesh, a quadrilateral mesh is frequently employed. However, while techniques, such as a bubble meshing method, are established for the automatic generation of a triangle mesh, as described in, for example, Japanese Unexamined Patent Publication Nos. Hei 7-230487 and Hei 8-315183, most of the techniques for the automatic generation of a quadrilateral mesh are not adequate for practical use. Therefore, many analysts generate a quadrilateral mesh by using a method that needs the expenditure of an enormous amount of human labor over several months to obtain CAD data for one car, for example.
While in the heat/fluid analysis field there are many demands for a mesh generation method for a three-dimensional geometric model, a method is desired for automatically generating a quadrilateral mesh (a hexahedral mesh in the three-dimensional space) that can be expanded to a three-dimensional model.
As mentioned above, there are numerous demands for an automatic generation method for a quadrilateral mesh. And not only an automatic generation method, but also a method that can cope with the following requests is desired.
(1) Less Distortion in a Generated Quadrangle Element
According to an analysis in computation mechanics, an extremely long element or an element having an extremely large (or small) angle adversely affects the result of the analysis. Ideally, therefore, all the quadrangle elements must be as close to being squares as is possible.
(2) Control of an Alignment Direction (arrangement direction) of Generated Quadrilateral Elements
According to an analysis in computation mechanics, it is preferable in many cases that elements are aligned in a direction of a physical value, such as stress, or along boundaries of a geometric model. Thus, it is desirable that a mesh is generated in which many elements are regularly arranged in a direction specified by a user.
(3) Control of a Distribution of Sizes of Elements
From the view of reduction in calculations, it is preferable that smaller mesh elements are generated in an important portion, and larger mesh elements are generated in a less important portion. However, when the size of the mesh element is suddenly changed, a T structure (a state where a node point of an adjacent element is placed on a chord of an element) is generated and adversely affects the analysis. It is, therefore, important that mesh elements in different sizes are distributed while it is ensured that the elements are connected by using shared node points and shared segments.
(4) Capability Handling Complicated Curved Shape as an Object
In models designed by CAD, there are various curved shapes, such as a trimmed curved shape obtained by removing one part of a curved region, or a very winding curved shape. It is desired that a quadrilateral mesh is automatically generated even for such a curved shape.
(5) Expansion into a Three-dimensional Geometric Model
An analysis employing the current computer mechanics develops a three-dimensional model from a two-dimensional geometric model in accordance with the enhancement of a calculation capacity. However, very few quadrilateral meshes (hexahedral meshes) are employed for a three-dimensional model. A meshing method is desired for employing the same operational principle to expand from a two-dimensional model to a three-dimensional model.
According to the above described bubble mesh method, bubbles are generated in a region of a geometric model and are moved to stable locations, and their center points are connected to generate a triangle mesh. Ordinarily, the following methods are employed to generate a quadrilateral mesh from such a triangle mesh.
(1) Method for Connecting two Adjacent Triangles to Form a Single Quadrangle.
The pairing of triangles is difficult, and when there is an odd number of triangles, an extra triangle remains. In addition, a quadrangle formed from two triangles is normally a parallelogram, and is neither a square nor a rectangle, both of which are appropriate shapes for a mesh.
(2) Method for Dividing a Triangle into Three Quadrangles.
With this method, it is difficult to generate an anisotropic quadrangle.
Either method is inadequate for the generation of a quadrangle performed while taking direction into account, and currently, this problem is resolved by applying a topologically heuristic rule. In addition, the above methods for changing a triangle mesh into a quadrilateral mesh are not practical, although theoretically with these methods the two-dimensional model could be expanded to the three-dimensional model.
Conventional methods for generating a quadrilateral mesh are as follows.
(1) Method for Employing Division of a Region by a User.
This method is a so-called "mapped mesh" method. A geometric model is divided into triangles or quadrangles in advance by a manual operation, such as the manipulation of a mouse, and the divided regions are further divided into lattices (see "Finite Element Mesh Generation Method: A Review And Classification," Ho-Le K., Computer Aided Designing, Vol. 20, No. 1, 1988, pp. 27-38). Since manual division of the regions is involved, this method does not satisfy the requirement for automatic generation. An enormous amount of manual labor would be required to generate a mesh for all the parts of a single car. In addition, since the results of the division, which is a pre-process, vary from user to user, the generated meshes differ, depending on the user.
(2) Method for Sequentially Generating Quadrilateral Elements from Boundaries of a Region.
This method is a so-called "advancing front mesh" generation method (see "A New Approach To Automated Quadrilateral Mesh Generation," Blacker T. D., Paving, International Journal For Numerical Methods In Engineering, Vol. 32, pp. 811-847, 1991). According to this method, a row of quadrilateral elements is generated along a boundary of the region and another row of quadrilateral elements is generated on the previous row in the region, and this process is repeated until the region is filled with quadrilateral elements. With this method, since the elements inside the region tend to be distorted, the above requirement (1) can not be satisfied. In addition, since the direction in which the elements are arranged is limited to a direction along the boundaries of the region, the above requirement (2) is not fully satisfied.